View source for Superconducting quantum computing

{{Short description|Quantum computing implementation}}
'''Superconducting quantum computing''' is a branch of [[Physics:Solid-state physics|solid state]] quantum computing that implements [[Physics:Superconductivity|superconducting]] [[Physics:Electronic circuit|electronic circuit]]s using superconducting qubits as artificial atoms, or [[Physics:Quantum dot|quantum dot]]s. For superconducting qubits, the two logic states are the [[Physics:Ground state|ground state]] and the [[Physics:Excited state|excited state]], denoted <math>|g\rangle \text{ and } |e\rangle</math>respectively.<ref name="docs.pennylane.ai" /> Research in superconducting quantum computing is conducted by companies such as [[Company:Google|Google]],<ref>{{cite journal |last1=Castelvecchi |first1=Davide |title=Quantum computers ready to leap out of the lab in 2017 |journal=Nature |date=5 January 2017 |volume=541 |issue=7635 |pages=9–10 |doi=10.1038/541009a|pmid=28054624 |bibcode=2017Natur.541....9C |s2cid=4447373 |doi-access=free }}</ref> [[Company:IBM|IBM]],<ref name="IBM">{{cite web |title=IBM Makes Quantum Computing Available on IBM Cloud |url=https://www-03.ibm.com/press/us/en/pressrelease/49661.wss |website=www-03.ibm.com |date=4 May 2016}}</ref> [[Company:IMEC|IMEC]],<ref>{{Cite web|url=https://www.imec-int.com/en/articles/imec-enters-the-race-to-unleash-quantum-computing-with-silicon-qubits|title=Imec enters the race to unleash quantum computing with silicon qubits|website=www.imec-int.com|language=en|access-date=2019-11-10}}</ref> [[Company:BBN Technologies|BBN Technologies]],<ref>Colm A. Ryan, Blake R. Johnson, Diego Ristè, Brian Donovan, Thomas A. Ohki, "Hardware for Dynamic Quantum Computing", arXiv:1704.08314v1</ref> [[Company:Rigetti Computing|Rigetti]],<ref>{{Cite news|url=https://www.hpcwire.com/2018/09/07/rigetti-launches-quantum-cloud-services-announces-1million-challenge/|title=Rigetti Launches Quantum Cloud Services, Announces $1Million Challenge|date=2018-09-07|work=HPCwire|access-date=2018-09-16|language=en-US}}</ref> and [[Company:Intel|Intel]].<ref>{{cite web |title=Intel Invests US$50 Million to Advance Quantum Computing {{!}} Intel Newsroom |url=https://newsroom.intel.com/news-releases/intel-invests-us50-million-to-advance-quantum-computing/ |website=Intel Newsroom}}</ref>  Many recently developed QPUs (quantum processing units, or quantum chips) use superconducting architecture.

{{As of|2016|May|df=US}}, up to 9 fully controllable [[Qubit|qubit]]s are demonstrated in the 1D [[Array programming|array]],<ref>{{cite journal |last1=Kelly |first1=J. |last2=Barends |first2=R. |last3=Fowler |first3=A. G. |last4=Megrant |first4=A. |last5=Jeffrey |first5=E. |last6=White |first6=T. C. |last7=Sank |first7=D. |last8=Mutus |first8=J. Y. |last9=Campbell |first9=B. |last10=Chen |first10=Yu |last11=Chen |first11=Z. |last12=Chiaro |first12=B. |last13=Dunsworth |first13=A. |last14=Hoi |first14=I.-C. |last15=Neill |first15=C. |last16=O’Malley |first16=P. J. J. |last17=Quintana |first17=C. |last18=Roushan |first18=P. |last19=Vainsencher |first19=A. |last20=Wenner |first20=J. |last21=Cleland |first21=A. N. |last22=Martinis |first22=John M. |title=State preservation by repetitive error detection in a superconducting quantum circuit |arxiv=1411.7403 |journal=Nature |date=4 March 2015 |volume=519 |issue=7541 |pages=66–69 |doi=10.1038/nature14270|pmid=25739628 |bibcode=2015Natur.519...66K |s2cid=3032369 }}</ref> and up to 16 in 2D architecture.<ref name="IBM" /> In October 2019, the Martinis group, partnered with [[Company:Google|Google]], published an article demonstrating novel [[Quantum supremacy|quantum supremacy]], using a chip composed of 53 superconducting qubits.<ref>{{cite journal |last1=Arute |first1=Frank |last2=Arya |first2=Kunal |last3=Babbush |first3=Ryan |last4=Bacon |first4=Dave |last5=Bardin |first5=Joseph C. |last6=Barends |first6=Rami |last7=Biswas |first7=Rupak |last8=Boixo |first8=Sergio |last9=Brandao |first9=Fernando G. S. L. |last10=Buell |first10=David A. |last11=Burkett |first11=Brian |last12=Chen |first12=Yu |last13=Chen |first13=Zijun |last14=Chiaro |first14=Ben |last15=Collins |first15=Roberto |last16=Courtney |first16=William |last17=Dunsworth |first17=Andrew |last18=Farhi |first18=Edward |last19=Foxen |first19=Brooks |last20=Fowler |first20=Austin |last21=Gidney |first21=Craig |last22=Giustina |first22=Marissa |last23=Graff |first23=Rob |last24=Guerin |first24=Keith |last25=Habegger |first25=Steve |last26=Harrigan |first26=Matthew P. |last27=Hartmann |first27=Michael J. |last28=Ho |first28=Alan |last29=Hoffmann |first29=Markus |last30=Huang |first30=Trent |last31=Humble |first31=Travis S. |last32=Isakov |first32=Sergei V. |last33=Jeffrey |first33=Evan |last34=Jiang |first34=Zhang |last35=Kafri |first35=Dvir |last36=Kechedzhi |first36=Kostyantyn |last37=Kelly |first37=Julian |last38=Klimov |first38=Paul V. |last39=Knysh |first39=Sergey |last40=Korotkov |first40=Alexander |last41=Kostritsa |first41=Fedor |last42=Landhuis |first42=David |last43=Lindmark |first43=Mike |last44=Lucero |first44=Erik |last45=Lyakh |first45=Dmitry |last46=Mandrà |first46=Salvatore |last47=McClean |first47=Jarrod R. |last48=McEwen |first48=Matthew |last49=Megrant |first49=Anthony |last50=Mi |first50=Xiao |last51=Michielsen |first51=Kristel |last52=Mohseni |first52=Masoud |last53=Mutus |first53=Josh |last54=Naaman |first54=Ofer |last55=Neeley |first55=Matthew |last56=Neill |first56=Charles |last57=Niu |first57=Murphy Yuezhen |last58=Ostby |first58=Eric |last59=Petukhov |first59=Andre |last60=Platt |first60=John C. |last61=Quintana |first61=Chris |last62=Rieffel |first62=Eleanor G. |last63=Roushan |first63=Pedram |last64=Rubin |first64=Nicholas C. |last65=Sank |first65=Daniel |last66=Satzinger |first66=Kevin J. |last67=Smelyanskiy |first67=Vadim |last68=Sung |first68=Kevin J. |last69=Trevithick |first69=Matthew D. |last70=Vainsencher |first70=Amit |last71=Villalonga |first71=Benjamin |last72=White |first72=Theodore |last73=Yao |first73=Z. Jamie |last74=Yeh |first74=Ping |last75=Zalcman |first75=Adam |last76=Neven |first76=Hartmut |last77=Martinis |first77=John M. |title=Quantum supremacy using a programmable superconducting processor |journal=Nature |date=October 2019 |volume=574 |issue=7779 |pages=505–510 |doi=10.1038/s41586-019-1666-5 |pmid=31645734 | arxiv=1910.11333|bibcode=2019Natur.574..505A |doi-access=free }}</ref>

==Background==
Classical [[Computation|computation]] models rely on physical implementations consistent with the laws of [[Physics:Classical mechanics|classical mechanics]].<ref>{{cite magazine |last1=Dayal |first1=Geeta |title=LEGO Turing Machine Is Simple, Yet Sublime |url=https://www.wired.com/2012/06/lego-turing-machine/ |magazine=WIRED}}</ref> Classical descriptions are accurate only for specific systems consisting of a relatively large number of atoms. A more general description of nature is given by [[Physics:Quantum mechanics|quantum mechanics]]. Quantum computation studies quantum phenomena applications beyond the scope of classical approximation, with the purpose of performing [[Quantum information|quantum information]] processing and communication. Various models of quantum computation exist, but the most popular models incorporate concepts of [[Qubit|qubit]]s and quantum gates (or gate-based superconducting quantum computing).

Superconductors are implemented due to the fact that at low temperatures they have infinite conductivity and zero resistance. Each qubit is built using semiconductor circuits with an [[Engineering:LC circuit|LC circuit]]: a capacitor and an inductor.{{Citation needed|date=January 2023}}

Superconducting capacitors and inductors are used to produce a resonant circuit that dissipates almost no energy, as heat  can disrupt quantum information. The superconducting resonant circuits are a class of artificial atoms that can be used as qubits. Theoretical and physical implementations of quantum circuits are widely different. Implementing a quantum circuit had its own set of challenges and must abide by [[DiVincenzo's criteria]], conditions proposed by theoretical physicist David P DiVincenzo,<ref name="qc-at-davis.github.io">{{Cite web |title=DiVincenzo's Criteria - Quantum Computing Codex |url=https://qc-at-davis.github.io/QCC/How-Quantum-Computing-Works/DiVincenzo%27s-Criteria/DiVincenzo%27s-Criteria.html#initialize-the-state-of-qubits-to-a-simple-fiducial-state |access-date=2022-12-13 |website=qc-at-davis.github.io}}</ref> which is set of criteria for the physical implementation of superconducting quantum computing, where the initial five criteria ensure that the quantum computer is in line with the postulates of quantum mechanics and the remaining two pertaining to the relaying of this information over a network.{{Citation needed|date=January 2023}}

We map the ground and excited states of these atoms to the 0 and 1 state as these are discrete and distinct energy values and therefore it is in line with the postulates of quantum mechanics. In such a construction however an electron can jump to multiple other energy states and not be confined to our excited state; therefore, it is imperative that the system be limited to be affected only by photons with energy difference required to jump from the ground state to the excited state.<ref name="pennylane.ai">{{Cite journal |title=Quantum computing with superconducting qubits — PennyLane |url=https://pennylane.ai/qml/demos/tutorial_sc_qubits.html |access-date=2022-12-13 |journal=Pennylane Demos|date=22 March 2022 |last1=Ballon |first1=Alvaro }}</ref> However, this leaves one major issue, we require uneven spacing between our energy levels to prevent photons with the same energy from causing transitions between neighboring pairs of states. Josephson junctions are superconducting elements with a nonlinear inductance, which is critically important for qubit implementation.<ref name="pennylane.ai"/> The use of this nonlinear element in the resonant superconducting circuit produces uneven spacings between the energy levels.{{Citation needed|date=January 2023}}

=== Qubits ===
A qubit is a generalization of a [[Bit|bit]] (a system with two possible [[Physics:Quantum state|states]]) capable of occupying a [[Physics:Quantum superposition|quantum superposition]] of both states. A quantum gate, on the other hand, is a generalization of a [[Engineering:Logic gate|logic gate]] describing the [[Transformation matrix|transformation]] of one or more qubits once a gate is applied given their initial state. Physical implementation of qubits and gates is challenging for the same reason that quantum phenomena are difficult to observe in everyday life given the minute scale on which they occur. One approach to achieving quantum computers is by implementing [[Physics:Superconductors|superconductors]] whereby quantum effects are [[Physics:Macroscopic scale|macroscopically]] observable, though at the price of extremely low operation [[Physics:Temperature|temperature]]s.

=== Superconductors ===
Unlike typical conductors, superconductors possess a [[Physics:Critical temperature|critical temperature]] at which resistivity plummets to nearly zero and conductivity is drastically increased. In superconductors, the basic charge carriers are pairs of [[Physics:Electron|electron]]s (known as [[Physics:Cooper pairs|Cooper pairs]]), rather than single [[Physics:Fermion|fermion]]s as found in typical conductors.<ref>{{Cite web |title=Cooper Pairs |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/coop.html }}</ref>  Cooper pairs are loosely bound and have an energy state lower than that of [[Physics:Fermi energy|Fermi Energy]]. Electrons forming Cooper pairs possess equal and opposite momentum and spin so that the total [[Physics:Spin|spin]] of the Cooper pair is an [[Integer|integer]] spin. Hence, Cooper pairs are [[Physics:Boson|boson]]s. Two such superconductors which have been used in superconducting qubit models are [[Chemistry:Niobium|niobium]] and [[Chemistry:Tantalum|tantalum]], both d-band superconductors.<ref>{{Cite journal |last=Shen |first=L. Y. L. |date=1972-02-01 |title=Superconductivity of Tantalum, Niobium and Lanthanum Studied by Electron Tunneling: Problems of Surface Contamination |url=https://aip.scitation.org/doi/abs/10.1063/1.2946195 |journal=AIP Conference Proceedings |volume=4 |issue=1 |pages=31–44 |doi=10.1063/1.2946195 |bibcode=1972AIPC....4...31S |issn=0094-243X}}</ref>

==== Bose-Einstein condensates ====
Once cooled to nearly [[Physics:Absolute zero|absolute zero]], a collection of bosons collapse into their lowest energy quantum state (the [[Physics:Ground state|ground state]]) to form a state of matter known as [[Physics:Bose–Einstein condensate|Bose–Einstein condensate]]. Unlike fermions, bosons may occupy the same quantum energy level (or [[Physics:Quantum state|quantum state]]) and do not obey the [[Physics:Pauli exclusion principle|Pauli exclusion principle]]. Classically, Bose-Einstein Condensate can be conceptualized as multiple particles occupying the same position in space and having equal [[Physics:Momentum|momentum]]. Because interactive forces between bosons are minimized, Bose-Einstein Condensates effectively act as a superconductor. Thus, superconductors are implemented in quantum computing because they possess both near infinite [[Physics:Conductivity (electrolytic)|conductivity]] and near zero [[Physics:Electrical resistance and conductance|resistance]]. The advantages of a superconductor over a typical conductor, then, are twofold in that superconductors can, in theory, transmit signals nearly instantaneously and run infinitely with no energy loss. The prospect of actualizing superconducting quantum computers becomes all the more promising considering [[Organization:NASA|NASA]]'s recent development of the [[Organization:Cold Atom Laboratory|Cold Atom Lab]] in outer space where Bose-Einstein Condensates are more readily achieved and sustained (without rapid dissipation) for longer periods of time without the constraints of [[Physics:Gravity|gravity]].<ref>{{Cite web |last=Greicius |first=Tony |date=2020-06-12 |title=NASA's Cold Atom Lab Takes One Giant Leap for Quantum Science |url=http://www.nasa.gov/feature/jpl/nasas-cold-atom-lab-takes-one-giant-leap-for-quantum-science |access-date=2022-12-11 |website=NASA}}</ref>

=== Electrical circuits ===
At each point of a superconducting electronic circuit (a network of [[Engineering:Electrical element|electrical element]]s), the condensate [[Wave function|wave function]] describing [[Physics:Electric charge|charge]] flow is well-defined by some [[Complex number|complex]] [[Physics:Probability amplitude|probability amplitude]]. In typical conductor electrical circuits, this same description is true for individual [[Physics:Charge carrier|charge carrier]]s except that the various wave functions are averaged in macroscopic analysis,  making it impossible to observe quantum effects. The condensate [[Wave function|wave function]] becomes useful in allowing design and measurement of macroscopic quantum effects. Similar to the discrete atomic [[Physics:Energy level|energy level]]s in the [[Physics:Bohr model|Bohr model]], only discrete numbers of [[Physics:Magnetic flux quantum|magnetic flux quanta]] can penetrate a superconducting loop. In both cases, [[Physics:Quantization|quantization]] results from complex [[Physics:Amplitude|amplitude]] [[Continuous function|continuity]]. Differing from microscopic implementations of quantum computers (such as [[Physics:Atom|atom]]s or [[Physics:Photon|photon]]s), parameters of superconducting circuits are designed by setting (classical) values to the electrical elements composing them such as by adjusting [[Physics:Capacitance|capacitance]] or [[Physics:Inductance|inductance]].

To obtain a quantum mechanical description of an electrical circuit, a few steps are required. Firstly, all electrical elements must be described by the condensate wave function amplitude and phase rather than by closely related macroscopic [[Physics:Electric current|current]] and [[Physics:Voltage|voltage]] descriptions used for classical circuits. For instance, the square of the wave function amplitude at any arbitrary point in space corresponds to the probability of finding a charge carrier there. Therefore, the squared amplitude corresponds to a classical charge distribution. The second requirement to obtain a quantum mechanical description of an electrical circuit is that generalized [[Kirchhoff's circuit laws]] are applied at every node of the circuit network to obtain the system's [[Physics:Equations of motion|equations of motion]]. Finally, these equations of motion must be reformulated to [[Lagrangian mechanics]] such that a [[Physics:Hamiltonian (quantum mechanics)|quantum Hamiltonian]] is derived describing the total energy of the system.
==Technology==

=== Manufacturing ===
Superconducting quantum computing devices are typically designed in the [[Engineering:Radio-frequency engineering|radio-frequency spectrum]], cooled in [[Physics:Dilution refrigerator|dilution refrigerator]]s below 15[[Kelvin|&nbsp;mK]] and addressed with conventional electronic instruments, e.g. [[Physics:Frequency synthesizer|frequency synthesizer]]s and [[Physics:Spectrum analyzer|spectrum analyzer]]s. Typical dimensions fall on the range of micrometers, with sub-micrometer resolution, allowing for the convenient design of a Hamiltonian system with well-established [[Engineering:Integrated circuit|integrated circuit]] technology. Manufacturing superconducting qubits follows a process involving lithography, depositing of metal, [[Chemistry:Etching|etching]], and controlled [[Chemistry:Redox|oxidation]] as described in.<ref name="Kjaergaard-2020">{{Cite journal |last1=Kjaergaard |first1=Morten |last2=Schwartz |first2=Mollie E. |last3=Braumüller |first3=Jochen |last4=Krantz |first4=Philip |last5=Wang |first5=Joel I.-Jan |last6=Gustavsson |first6=Simon |last7=Oliver |first7=William D. |date=2020-03-10 |title=Superconducting Qubits: Current State of Play |journal=Annual Review of Condensed Matter Physics |volume=11 |issue=1 |pages=369–395 |doi=10.1146/annurev-conmatphys-031119-050605 |arxiv=1905.13641 |bibcode=2020ARCMP..11..369K |s2cid=173188891 |issn=1947-5454}}</ref> Manufacturers continue to improve the lifetime of superconducting qubits and have made significant improvements since the early 2000s.{{r|Kjaergaard-2020|page=4|comment=chart on the right on page 4}}

=== Josephson junctions ===
[[File:Single josephson junction.svg|thumb|A single Josephson junction where C is a thin layer of insulator and A & B are (superconducting) currents with nonequivalent wave functions]]
One distinguishable attribute of superconducting quantum circuits is the use of Josephson junctions. Josephson junctions are an [[Engineering:Electrical element|electrical element]] which does not exist in [[Physics:Superconductivity#Superconducting phase transition|normal conductors]]. Recall that a [[Physics:Electrical junction|junction]] is a weak connection between two leads of wire (in this case a superconductive wire) on either side of a thin layer of [[Physics:Insulator (electricity)|insulator]] material only a few [[Physics:Atom|atom]]s thick, usually implemented using [[Engineering:Niemeyer–Dolan technique|shadow evaporation]] technique. The resulting Josephson junction device exhibits the [[Physics:Josephson effect|Josephson Effect]] whereby the junction produces a [[Physics:Supercurrent|supercurrent]]. An image of a single Josephson junction is shown to the right. The condensate wave function on the two sides of the junction are weakly correlated, meaning that they are allowed to have different superconducting phases. This distinction of [[Linearity|nonlinearity]] contrasts continuous superconducting wire for which the wave function across the junction must be [[Continuous function|continuous]]. Current flow through the junction occurs by quantum tunneling, seeming to instantaneously "tunnel" from one side of the junction to the other. This tunneling phenomenon is unique to quantum systems. Thus, quantum tunneling is used to create nonlinear inductance, essential for qubit design as it allows a design of [[Physics:Anharmonicity|anharmonic oscillators]] for which energy levels are discretized (or [[Virtual memory compression#Compression using quantization|quantize]]d) with nonuniform spacing between energy levels, denoted <math>\Delta E</math>.<ref name="docs.pennylane.ai" /> In contrast, the [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]] ''cannot'' be used as a qubit as there is no way to address only two of its states, given that the spacing between every energy level and the next is exactly the same.

==Qubit archetypes==
The three primary superconducting qubit archetypes are the [[Phase qubit|phase]], [[Charge qubit|charge]] and [[Flux qubit|flux]] qubit. Many hybridizations of these archetypes exist including the fluxonium,<ref>{{cite journal|last1=Manucharyan|first1=V. E.|last2=Koch|first2=J.|last3=Glazman|first3=L. I.|last4=Devoret|first4=M. H.|title=Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets|arxiv=0906.0831|journal=Science|date=1 October 2009|volume=326|issue=5949|pages=113–116|doi=10.1126/science.1175552|pmid=19797655 |bibcode=2009Sci...326..113M|s2cid=17645288}}</ref> [[Transmon|transmon]],<ref>{{cite journal |arxiv=0812.1865 |last1=Houck |first1=A. A. |last2=Koch |first2=Jens |last3=Devoret |first3=M. H. |last4=Girvin |first4=S. M. |last5=Schoelkopf |first5=R. J. |title=Life after charge noise: recent results with transmon qubits |journal=Quantum Information Processing |date=11 February 2009 |volume=8 |issue=2–3 |pages=105–115 |doi=10.1007/s11128-009-0100-6|s2cid=27305073 }}</ref> Xmon,<ref>{{cite journal |last1=Barends |first1=R. |last2=Kelly |first2=J. |last3=Megrant |first3=A. |last4=Sank |first4=D. |last5=Jeffrey |first5=E. |last6=Chen |first6=Y. |last7=Yin |first7=Y. |last8=Chiaro |first8=B. |last9=Mutus |first9=J. |last10=Neill |first10=C. |last11=O’Malley |first11=P. |last12=Roushan |first12=P. |last13=Wenner |first13=J. |last14=White |first14=T. C. |last15=Cleland |first15=A. N. |last16=Martinis |first16=John M. |title=Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits |arxiv=1304.2322 |journal=Physical Review Letters |date=22 August 2013 |volume=111 |issue=8 |pages=080502 |doi=10.1103/PhysRevLett.111.080502|pmid=24010421 |bibcode=2013PhRvL.111h0502B |s2cid=27081288 }}</ref> and quantronium.<ref>{{cite journal |last1=Metcalfe |first1=M. |last2=Boaknin |first2=E. |last3=Manucharyan |first3=V. |last4=Vijay |first4=R. |last5=Siddiqi |first5=I. |last6=Rigetti |first6=C. |last7=Frunzio |first7=L. |last8=Schoelkopf |first8=R. J. |last9=Devoret |first9=M. H. |title=Measuring the decoherence of a quantronium qubit with the cavity bifurcation amplifier |arxiv=0706.0765 |journal=Physical Review B |date=21 November 2007 |volume=76 |issue=17 |pages=174516 |doi=10.1103/PhysRevB.76.174516|bibcode=2007PhRvB..76q4516M |s2cid=19088840 }}</ref> For any qubit implementation the logical quantum states <math>\{|0\rangle,|1\rangle\}</math> are [[Map (mathematics)|mapped]] to different states of the physical system (typically to discrete [[Physics:Energy level|energy level]]s or their [[Physics:Quantum superposition|quantum superposition]]s). Each of the three archetypes possess a distinct range of Josephson energy to charging energy ratio. Josephson energy refers to the energy stored in Josephson junctions when current passes through, and charging energy is the energy required for one Cooper pair to charge the junction's total capacitance.<ref name="Martinis-2004">{{Cite journal |last1=Martinis |first1=John M. |last2=Osborne |first2=Kevin |date=2004-02-16 |title=Superconducting Qubits and the Physics of Josephson Junctions |arxiv=cond-mat/0402415 |bibcode=2004cond.mat..2415M }}</ref> Josephson energy can be written as  [[File:Energy scales for qubits.png|thumb|A graph of various superconducting qubit archetypes by their Josephson energy to charging energy ratio with a legend on the right.<ref name="Hyyppä-2022">{{Cite journal |last1=Hyyppä |first1=Eric |last2=Kundu |first2=Suman |last3=Chan |first3=Chun Fai |last4=Gunyhó |first4=András |last5=Hotari |first5=Juho |last6=Janzso |first6=David |last7=Juliusson |first7=Kristinn |last8=Kiuru |first8=Olavi |last9=Kotilahti |first9=Janne |last10=Landra |first10=Alessandro |last11=Liu |first11=Wei |last12=Marxer |first12=Fabian |last13=Mäkinen |first13=Akseli |last14=Orgiazzi |first14=Jean-Luc |last15=Palma |first15=Mario |date=2022-11-12 |title=Unimon qubit |journal=Nature Communications |language=en |volume=13 |issue=1 |pages=6895 |doi=10.1038/s41467-022-34614-w |pmid=36371435 |pmc=9653402 |arxiv=2203.05896 |bibcode=2022NatCo..13.6895H |issn=2041-1723}}</ref> The top left graphic illustrates a unimon electrical circuit.<ref name="Hyyppä-2022" />]]<math>U_j = - \frac{I_0 \Phi_0}{2 \pi} cos \delta</math>,

where <math>I_0</math> is the critical current parameter of the Josephson junction, <math>\Phi_0 = \frac{h}{2e}</math> is (superconducting) [[Physics:Magnetic flux quantum|flux quantum]], and <math>\delta</math> is the [[Physics:Phase (waves)|phase difference]] across the junction.<ref name="Martinis-2004" /> Notice that the term <math>cos \delta</math> indicates nonlinearity of the Josephson junction.<ref name="Martinis-2004" /> Charge energy is written as

<math>E_C = \frac{e^2}{2C}</math>,

where <math>C</math> is the junction's capacitance and <math>e</math> is electron charge.<ref name="Martinis-2004" /> Of the three archetypes, phase qubits allow the most of Cooper pairs to tunnel through the junction, followed by flux qubits, and charge qubits allow the fewest.

=== Phase qubit ===
The phase qubit possesses a Josephson to charge energy ratio on the order of magnitude <math>10^6</math>. For phase qubits, energy levels correspond to different quantum charge oscillation [[Physics:Amplitude|amplitude]]s across a Josephson junction, where charge and [[Physics:Ginzburg–Landau theory|phase]] are analogous to momentum and position respectively as analogous to a [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]]. Note that in this context phase is the complex argument of the superconducting wave function (also known as the superconducting [[Physics:Phase transition|order parameter]]), not the phase between the different states of the qubit.[[File:Fluxonium Qubit.png|thumb|The left-most image shows a fluxonium superconducting loop consisting of a collection of larger area Josephson junctions and one smaller area Josephson junction, as shown by an electron microscope.<ref name="Cottet 6383">{{Cite journal |last1=Cottet |first1=Nathanaël |last2=Xiong |first2=Haonan |last3=Nguyen |first3=Long B. |last4=Lin |first4=Yen-Hsiang |last5=Manucharyan |first5=Vladimir E. |date=2021-11-04 |title=Electron shelving of a superconducting artificial atom |journal=Nature Communications |language=en |volume=12 |issue=1 |pages=6383 |doi=10.1038/s41467-021-26686-x |pmid=34737313 |pmc=8569191 |arxiv=2008.02423 |bibcode=2021NatCo..12.6383C |issn=2041-1723}}</ref> The top right image depicts fluxonium circuit components, and the bottom right image depicts a smaller area Josephson junction.<ref name="Cottet 6383"/>]]

=== Flux qubit ===
The flux qubit (also known as a persistent-current qubit) possesses a Josephson to charging energy ratio on the order of magnitude <math>10</math>. For flux qubits, the energy levels correspond to different [[Integer|integer]] numbers of magnetic flux quanta trapped in a superconducting ring.

==== Fluxonium ====
Fluxonium qubits are a specific type of flux qubit whose Josephson junction is shunted by a linear inductor of <math> E_{J} \gg E_{L} </math> where <math>E_L = (\hbar/2e)^2 / L </math>.<ref name="Nguyen-2019" /> In practice, the linear inductor is usually implemented by a Josephson junction array that is composed of a large number (can be often <math> N > 100 </math>) of large-sized Josephson junctions connected in a series. Under this condition, the Hamiltonian of a fluxonium can be written as the following:

<math>H = 4 E_C n^2 + \frac{1}{2} E_L \phi^2 - E_J \cos(\phi - \phi_{ext})</math>.

One important advantage of a fluxonium qubit is the longer [[Physics:Coherence#Quantum coherence|qubit lifetime]] at the half flux sweet spot, which can exceed 1 millisecond.<ref>{{Cite web |last1=Science |first1=The National University of |last2=MISIS |first2=Technology |title=Fluxonium qubits bring the creation of a quantum computer closer |url=https://phys.org/news/2022-11-fluxonium-qubits-creation-quantum-closer.html |access-date=2022-12-12 |website=phys.org |language=en}}</ref><ref name="Nguyen-2019">{{Cite journal |last1=Nguyen |first1=Long B. |last2=Lin |first2=Yen-Hsiang |last3=Somoroff |first3=Aaron |last4=Mencia |first4=Raymond |last5=Grabon |first5=Nicholas |last6=Manucharyan |first6=Vladimir E. |date=2019-11-25 |title=High-Coherence Fluxonium Qubit |url=https://link.aps.org/doi/10.1103/PhysRevX.9.041041 |journal=Physical Review X |language=en |volume=9 |issue=4 |pages=041041 |doi=10.1103/PhysRevX.9.041041 |arxiv=1810.11006 |bibcode=2019PhRvX...9d1041N |s2cid=53499609 |issn=2160-3308}}</ref>

=== Charge qubit ===
The charge qubit, also known as the Cooper pair box, possesses a Josephson to charging energy ratio on the order of magnitude <math>< 1</math>. For charge qubits, different energy levels correspond to an integer number of [[Physics:Cooper pair|Cooper pair]]s on a superconducting island (a small superconducting area with a controllable number of charge carriers).<ref>{{Cite web |date=2019-06-06 |title=Superconducting qubits – on islands, charge qubits and the transmon |url=https://leftasexercise.com/2019/06/06/superconducting-qubits-on-islands-charge-qubits-and-the-transmon/ |access-date=2022-12-12 |website=LeftAsExercise |language=en}}</ref> Indeed, the first experimentally realized qubit was the Cooper pair box, achieved in 1999.<ref>{{Cite journal |last=Wendin |first=G. |date=2017-10-01 |title=Quantum information processing with superconducting circuits: a review |journal=Reports on Progress in Physics |volume=80 |issue=10 |pages=106001 |doi=10.1088/1361-6633/aa7e1a |pmid=28682303 |arxiv=1610.02208 |bibcode=2017RPPh...80j6001W |s2cid=3940479 |issn=0034-4885}}</ref>  [[File:4 Qubit, 4 Bus, 4 Resonator IBM Device (Jay M. Gambetta, Jerry M. Chow, and Matthias Steffen, 2017).png|thumb|upright=0.8|A device consisting of four superconducting [[Transmon|transmon]] qubits, four [[Quantum bus|quantum bu]]ses, and four readout resonators fabricated by IBM and published in npj Quantum Information in January 2017<ref>{{cite journal | last1 = Gambetta | first1 = J. M. |  last2 = Chow | first2 = J. M. | last3 = Steffen | first3 = M. | year = 2017 | title = Building logical qubits in a superconducting quantum computing system | journal = npj Quantum Information | volume = 3 | issue = 1| page = 2 | doi = 10.1038/s41534-016-0004-0 | bibcode = 2017npjQI...3....2G | doi-access = free | arxiv = 1510.04375 }}</ref>|alt=]]

==== Transmon ====
Transmons are a special type of qubit with a [[Engineering:Shunt (electrical)|shunted]] capacitor specifically designed to mitigate [[Physics:Quantum noise|noise]]. The transmon qubit model was based on the Cooper pair box<ref>{{Cite journal |last1=Roth |first1=Thomas E. |last2=Ma |first2=Ruichao |last3=Chew |first3=Weng C. |date=2021-06-21 |title=An Introduction to the Transmon Qubit for Electromagnetic Engineers |arxiv=2106.11352 }}</ref> (illustrated in the table above in row one column one). It was also the first qubit to demonstrate [[Quantum supremacy|quantum supremacy]].<ref>{{Cite journal |last1=Kjaergaard |first1=Morten |last2=Schwartz |first2=Mollie E. |last3=Braumüller |first3=Jochen |last4=Krantz |first4=Philip |last5=Wang |first5=Joel I.-J. |last6=Gustavsson |first6=Simon |last7=Oliver |first7=William D. |date=2020-03-10 |title=Superconducting Qubits: Current State of Play |url=https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-031119-050605 |journal=Annual Review of Condensed Matter Physics |language=en |volume=11 |issue=1 |pages=369–395 |doi=10.1146/annurev-conmatphys-031119-050605 |arxiv=1905.13641 |bibcode=2020ARCMP..11..369K |s2cid=173188891 |issn=1947-5454}}</ref> The increased ratio of Josephson to charge energy mitigates noise. Two transmons can be coupled using a [[Engineering:Capacitive coupling|coupling capacitor]].<ref name="docs.pennylane.ai" /> For this 2-qubit system the Hamiltonian is written

<math>\hat{H} = \frac{\hbar J}{2} (\sigma_{1}^{x} \sigma_{2}^{x} + \sigma_{1}^{y} \sigma_{2}^{y})</math>,

where <math>J</math> is [[Physics:Current density|current density]] and <math>\sigma</math> is [[Physics:Charge density|surface charge density]].<ref name="docs.pennylane.ai" />

==== Xmon ====
The Xmon is very similar in design to a transmon in that it originated based on the planar transmon model.<ref>{{Cite journal |last1=Shim |first1=Yun-Pil |last2=Tahan |first2=Charles |date=2016-03-17 |title=Semiconductor-inspired design principles for superconducting quantum computing |journal=Nature Communications |language=en |volume=7 |issue=1 |pages=11059 |doi=10.1038/ncomms11059 |pmid=26983379 |pmc=4800439 |arxiv=1507.07923 |bibcode=2016NatCo...711059S |issn=2041-1723}}</ref> An Xmon is essentially a tunable transmon. The major distinguishing difference between transmon and Xmon qubits is the Xmon qubits is grounded with one of its capacitor pads.<ref>{{Cite journal |last1=Wang |first1=Chenlu |last2=Li |first2=Xuegang |last3=Xu |first3=Huikai |last4=Li |first4=Zhiyuan |last5=Wang |first5=Junhua |last6=Yang |first6=Zhen |last7=Mi |first7=Zhenyu |last8=Liang |first8=Xuehui |last9=Su |first9=Tang |last10=Yang |first10=Chuhong |last11=Wang |first11=Guangyue |last12=Wang |first12=Wenyan |last13=Li |first13=Yongchao |last14=Chen |first14=Mo |last15=Li |first15=Chengyao |date=2022-01-13 |title=Towards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds |url=https://www.nature.com/articles/s41534-021-00510-2 |journal=npj Quantum Information |language=en |volume=8 |issue=1 |page=3 |doi=10.1038/s41534-021-00510-2 |arxiv=2105.09890 |bibcode=2022npjQI...8....3W |s2cid=245950831 |issn=2056-6387}}</ref>

==== Gatemon ====
Another variation of the transmon qubit is the Gatemon. Like the Xmon, the Gatemon is a tunable variation of the transmon. The Gatemon is tunable via [[Physics:Threshold voltage|gate voltage]].  [[File:Chip unimon.png|thumb|Superconducting circuit consisting of 3 Unimons (blue), each connected to resonators (red), drive lines (green), and joint probe lines (yellow)<ref>{{Cite journal |title=Fig. 1: Unimon qubit and its measurement setup. {{!}} Nature Communications |url=https://www.nature.com/articles/s41467-022-34614-w/figures/1 |language=en}}</ref>]]

=== Unimon ===
In 2022 researchers from IQM Quantum Computers, Aalto University, and VTT Technical Research Centre of Finland discovered a novel superconducting qubit known as the Unimon.<ref>{{Cite web |title=Unimon: A new qubit to boost quantum computers from IQM {{!}} IQM |url=https://www.meetiqm.com/articles/press-releases/iqm-unimon-qubit/ |access-date=2022-12-12 |website=www.meetiqm.com |language=en}}</ref> A relatively simple qubit, the Unimon consists of a single Josephson junction shunted by a linear inductor (possessing an inductance not depending on current) inside a (superconducting) [[Physics:Resonator|resonator]].<ref name="Buchanan-2022">{{Cite journal |last=Buchanan |first=Mark |date=2022-12-08 |title=Meet the Unimon, the New Qubit on the Block |url=https://physics.aps.org/articles/v15/191 |journal=Physics |language=en |volume=15|page=191 |doi=10.1103/Physics.15.191 |bibcode=2022PhyOJ..15..191B |s2cid=257514449 |doi-access=free }}</ref> Unimons have increased anharmocity and display faster operation time resulting in lower susceptibility to noise errors.<ref name="Buchanan-2022" /> In addition to increased anharmocity, other advantages Unimon qubit include decreased susceptibility to flux noise and complete insensitivity to dc charge noise.<ref name="Hyyppä-2022" />    
{| class="wikitable"
|+Superconducting Qubit Archetypes
<ref name="ShortRev">{{cite arXiv|last1=Devoret|first1=M. H.|last2=Wallraff|first2=A.|last3=Martinis|first3=J. M.|title=Superconducting Qubits: A Short Review|eprint=cond-mat/0411174|date=6 November 2004}}</ref>
|-
! {{diagonal split header|Aspect|Type}}
! [[Charge qubit]] !! RF-SQUID qubit (prototype of the Flux Qubit) !! Phase qubit
|-
!Circuit
|[[File:Cooper pair box circuit.png|thumb|Charge qubit circuit. A superconducting island (encircled with a dashed line) is defined between the leads of a capacitor with capacitance <math>C</math> and a Josephson junction with energy <math>E_J</math> biased by voltage <math>U</math>. ]]

||
[[File:Flux qubit circuit.svg|thumb|Flux qubit circuit. A superconducting loop with inductance <math>L</math> is interrupted by a junction with Josephson energy <math>E_J</math>. Bias flux <math>\Phi</math> is induced by a flux line with current <math>I_0</math>. ]]

||
[[File:PhaseQBcirc.svg|thumb|Phase qubit circuit. A Josephson junction with energy parameter <math>E_J</math> is biased by current <math>I_0</math>. ]]

|-
! [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]]
|
<math>
H=E_C(N-N_g)^2-E_J\cos\phi
</math>

In this case <math>N</math> is the number of [[Physics:Cooper pairs|Cooper pairs]] to [[Physics:Quantum tunnelling|tunnel]] through the junction, <math>N_g=CV_0/2e</math> is the charge on the [[Engineering:Capacitor|capacitor]] in units of Cooper pairs number, <math>E_C=(2e)^2/2(C_J+C)</math> is the charging energy associated with both capacitance <math>C</math> and Josephson junction capacitance <math>C_J</math>.  
||
<math>
H=\frac{q^2}{2C_J}+\left(\frac{\Phi_0}{2\pi}\right)^2\frac{\phi^2}{2L}-E_J\cos\left[\phi-\Phi\frac{2\pi}{\Phi_0}\right] 
</math>

Note that <math>\phi</math> is only allowed to take values greater than <math>2\pi</math> and is alternatively defined as the time integral of voltage along inductance <math>L</math>.
||
<math>
H=\frac{(2e)^2}{2C_J}q^2-I_0\frac{\Phi_0}{2\pi}\phi-E_J\cos\phi
</math>Here <math>\Phi_0</math> is magnetic flux quantum.
|-
! [[Physics:Potential energy|Potential]]
|[[File:Charge qubit potential.svg|thumb|<math>U = -E_J\cos\phi</math> . Bias voltage is set such that <math>N_g=\frac{1}{2}</math>, minimizing the energy gap between <math>|0\rangle</math> and <math>|1\rangle</math>, consequently distinguishing the gap from other energy gaps (e.g. gap between <math>|1\rangle</math> and <math>|2\rangle</math>). The difference in gaps allows addressing transitions from <math>|0\rangle</math> to <math>|1\rangle</math> and vice versa only, without populating other states.  ]]

||
[[File:Flux qubit potential.svg|thumb|<math>U = \left(\frac{\Phi_0}{2\pi}\right)^2\frac{\phi^2}{2L}-E_J\cos\left[\phi-\Phi\frac{2\pi}{\Phi_0}\right]</math> Bias flux is <math>\Phi=\Phi_0/2</math>. Different wells correspond to a distinct number of flux quanta trapped in the superconducting loops. The two lower states correspond to a symmetrical and anti-symmetrical superposition of zero or single trapped flux quanta, sometimes denoted as clockwise and counterclockwise loop current states: <math>|0\rangle=\left[|\circlearrowleft\rangle+|\circlearrowright\rangle\right]/\sqrt{2}</math> and <math>|1\rangle=\left[|\circlearrowleft\rangle-|\circlearrowright\rangle\right]/\sqrt{2}</math>. ]]
 
||
[[File:Phase qubit potential.svg|thumb|<math>U = -I_0\frac{\Phi_0}{2\pi}\phi-E_J\cos\phi</math>, also known as "washboard" potential. Bias current is adjusted to allow wells shallow enough to contain exactly two localized wave functions. A slight increase in bias current causes a selective "spill" of higher energy state (<math>|1\rangle</math>), expressed with a measurable voltage spike (a mechanism commonly used for phase qubit measurement). ]]

|}
In the table above, the three superconducting qubit archetypes are reviewed. In the first row, the qubit's electrical circuit diagram is presented. The second row depicts a quantum Hamiltonian derived from the circuit. Generally, the Hamiltonian is the sum of the system's [[Physics:Kinetic energy|kinetic]] and [[Physics:Potential energy|potential]] energy components (analogous to a particle in a [[Physics:Potential well|potential well]]). For the Hamiltonians denoted, <math>\phi</math> is the superconducting wave function phase difference across the junction, <math>C_J</math> is the capacitance associated with the Josephson junction, and <math>q</math> is the charge on the junction capacitance. For each potential depicted, only solid wave functions are used for computation. The qubit potential is indicated by a thick red line, and schematic wave function solutions are depicted by thin lines, lifted to their appropriate energy level for clarity.

Note that particle mass corresponds to an [[Inverse function|inverse function]] of the circuit capacitance and that the shape of the potential is governed by regular [[Engineering:Inductor|inductor]]s and Josephson junctions. Schematic wave solutions in the third row of the table show the complex amplitude of the phase variable. Specifically, if a qubit's phase is measured while the qubit occupies a particular state, there is a non-zero probability of measuring a specific value ''only'' where the depicted wave function oscillates. All three rows are essentially different presentations of the same physical system.

==Single qubits==
The [[Hertz|GHz]] energy gap between energy levels of a superconducting qubit is designed to be compatible with available electronic equipment, due to the [[Physics:Terahertz gap|terahertz gap]] (lack of equipment in the higher [[Frequency band|frequency band]]). The [[Physics:BCS theory#Implications|superconductor energy gap]] implies a top limit of operation below ~1THz beyond which Cooper pairs break, so energy level separation cannot be too high. On the other hand, energy level separation cannot be too small due to cooling considerations: a temperature of 1&nbsp;K implies [[Physics:Thermal fluctuations|energy fluctuations]] of 20&nbsp;GHz. Temperatures of tens of millikelvins are achieved in [[Physics:Dilution refrigerator|dilution refrigerator]]s and allow qubit operation at a ~5&nbsp;GHz energy level separation. Qubit energy level separation is frequently adjusted by controlling a dedicated [[Engineering:Biasing|bias current]] line, providing a "knob" to fine tune the qubit parameters.

===Single qubit gates===
[[File:Bloch Sphere.svg|thumb|A depiction of the Bloch sphere]]
A single qubit gate is achieved by rotation in the [[Physics:Bloch sphere|Bloch sphere]]. Rotations between different energy levels of a single qubit are induced by [[Physics:Microwave|microwave]] pulses sent to an [[Engineering:Antenna (radio)|antenna]] or [[Engineering:Transmission line|transmission line]] coupled to the qubit with a [[Physics:Frequency|frequency]] resonant with the energy separation between levels. Individual qubits may be addressed by a dedicated [[Engineering:Transmission line|transmission line]] or by a shared one if the other qubits are off [[Physics:Resonance|resonance]]. The [[Rotation around a fixed axis|axis of rotation]] is set by [[Quadrature amplitude modulation|quadrature amplitude modulation]] of microwave pulse, while pulse length determines the [[Angle of rotation|angle of rotation]].<ref name="oneQBgate">{{cite journal |last1=Motzoi |first1=F. |last2=Gambetta |first2=J. M. |last3=Rebentrost |first3=P. |last4=Wilhelm |first4=F. K. |title=Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits |arxiv=0901.0534 |journal=Physical Review Letters |date=8 September 2009 |volume=103 |issue=11 |pages=110501 |doi=10.1103/PhysRevLett.103.110501|pmid=19792356 |bibcode=2009PhRvL.103k0501M |s2cid=7288207 }}</ref>

More formally (following the notation of <ref name=oneQBgate/>) for a driving signal

<math>\mathcal{E}(t)=\mathcal{E}^x(t)\cos(\omega_d t)+\mathcal{E}^y(t)\sin(\omega_d t)</math>

of frequency <math>\omega_d</math>, a driven qubit Hamiltonian in a [[Physics:Rotating wave approximation|rotating wave approximation]] is

<math>H^R/\hbar=(\omega-\omega_d)|1\rangle\langle 1|+\frac{\mathcal{E}^x(t)}{2}\sigma_x+\frac{\mathcal{E}^y(t)}{2}\sigma_y</math>,

where <math>\omega</math> is the qubit resonance and <math>\sigma_x,\sigma_y</math> are [[Pauli matrices]].

To implement a rotation about the <math>X</math> axis, one can set <math>\mathcal{E}^y(t)=0</math> and apply a microwave pulse at frequency <math>\omega_d=\omega</math> for time <math>t_g</math>. The resulting transformation is

<math>U_x=\exp\left\{-\frac{i}{\hbar}\int_0^{t_g}H^R dt\right\}=\exp\left\{-i\int_0^{t_g}\mathcal{E}^x(t) dt\cdot\sigma_x/2\right\}</math>.

This is exactly the [[Physics:Rotation operator (quantum mechanics)|rotation operator]] <math>R_X(\theta)</math> by angle <math>\theta=\int_0^{t_g}\mathcal{E}^x(t) dt</math> about the <math>X</math> axis in the Bloch sphere. A rotation about the <math>Y</math> axis can be implemented in a similar way. Showing the two rotation operators is sufficient for satisfying universality as every single qubit unitary operator <math>U</math> may be presented as <math>U=R_X(\theta_1)R_Y(\theta_2)R_X(\theta_3)</math> (up to a global [[Phase factor|phase]] which is physically inconsequential) by a procedure known as the <math>X-Y</math> decomposition.<ref>{{cite book |last1=Chuang |first1=Michael A. Nielsen & Isaac L. |title=Quantum computation and quantum information |url=https://archive.org/details/quantumcomputati00niel |url-access=limited |date=2010 |publisher=Cambridge University Press |location=Cambridge |isbn=978-1-107-00217-3 |pages=[https://archive.org/details/quantumcomputati00niel/page/n207 174]–176 |edition=10th anniversary}}</ref>  Setting <math>\int_0^{t_g}\mathcal{E}^x(t) dt=\pi</math> results in the transformation

<math>U_x=\exp\left\{-i\int_0^{t_g}\mathcal{E}^x(t) dt\cdot\sigma_x/2\right\}=e^{-i\pi\sigma_x/2}=-i\sigma_x</math>

up to the global phase <math>-i</math> and is known as the NOT gate.

==Coupling qubits==
The ability to couple qubits is essential for implementing 2-qubit [[Quantum logic gate|gates]]. Coupling two qubits can be achieved by connecting both to an intermediate electrical coupling circuit. The circuit may be either a fixed element (such as a capacitor) or be controllable (like the [[Physics:SQUID#DC SQUID|DC-SQUID]]). In the first case, [[Engineering:Decoupling (electronics)|decoupling]] qubits during the time the gate is switched off is achieved by tuning qubits out of resonance one from another, making the energy gaps between their computational states different.<ref>{{cite book |last1=Rigetti |first1=Chad Tyler |title=Quantum gates for superconducting qubits |page=21 |bibcode=2009PhDT........50R |isbn=9781109198874|year=2009 }}</ref> This approach is inherently limited to nearest-neighbor coupling since a physical electrical circuit must be laid out between connected qubits. Notably, [[Company:D-Wave Systems|D-Wave Systems]]' nearest-neighbor coupling achieves a highly connected [[Unit cell|unit cell]] of 8 qubits in Chimera graph configuration. [[Quantum algorithm]]s typically require coupling between arbitrary qubits. Consequently, multiple swap operations are necessary, limiting the length  of quantum computation possible before processor [[Physics:Quantum decoherence|decoherence]].

=== Quantum bus ===
Another method of coupling two or more qubits is by way of a [[Quantum bus|quantum bus]], by pairing qubits to this intermediate. A quantum bus is often implemented as a [[Physics:Microwave cavity|microwave cavity]] modeled by a quantum harmonic oscillator. Coupled qubits may be brought in and out of resonance with the bus and with each other, eliminating the nearest-neighbor limitation. Formalism describing coupling is [[Cavity quantum electrodynamics|cavity quantum electrodynamics]]. In cavity quantum electrodynamics, qubits are analogous to atoms interacting with an [[Physics:Optical cavity|optical photon cavity]] with a difference of GHz (rather than the THz regime of electromagnetic radiation). Resonant excitation exchange among these artificial atoms is potentially useful for direct implementation of multi-qubit gates.<ref name="Khazali-2020">{{Cite journal|last1=Khazali|first1=Mohammadsadegh|last2=Mølmer|first2=Klaus|date=2020-06-11|title=Fast Multiqubit Gates by Adiabatic Evolution in Interacting Excited-State Manifolds of Rydberg Atoms and Superconducting Circuits|journal=Physical Review X|language=en|volume=10|issue=2|pages=021054|doi=10.1103/PhysRevX.10.021054|arxiv=2006.07035 |bibcode=2020PhRvX..10b1054K |issn=2160-3308|doi-access=free}}</ref> Following the dark state [[Manifold|manifold]], the Khazali-Mølmer scheme<ref name="Khazali-2020" /> performs complex multi-qubit operations in a single step, providing a substantial shortcut to the conventional circuit model.

===Cross resonant gate===
One popular gating mechanism uses two qubits and a bus, each tuned to different energy level separations. Applying microwave excitation to the first qubit, with a frequency resonant with the second qubit, causes a <math>\sigma_x</math> rotation of the second qubit. Rotation direction depends on the state of the first qubit, allowing a controlled phase gate construction.<ref name=CRgate>{{cite journal |last1=Chow |first1=Jerry M. |last2=Córcoles |first2=A. D. |last3=Gambetta |first3=Jay M. |last4=Rigetti |first4=Chad |last5=Johnson |first5=B. R. |last6=Smolin |first6=John A. |last7=Rozen |first7=J. R. |last8=Keefe |first8=George A. |last9=Rothwell |first9=Mary B. |last10=Ketchen |first10=Mark B. |last11=Steffen |first11=M. |title=Simple All-Microwave Entangling Gate for Fixed-Frequency Superconducting Qubits |arxiv=1106.0553 |journal=Physical Review Letters |date=17 August 2011 |volume=107 |issue=8 |pages=080502 |doi=10.1103/PhysRevLett.107.080502|pmid=21929152 |bibcode=2011PhRvL.107h0502C |s2cid=9302474 }}</ref>

Following the notation of,<ref name=CRgate/> the drive Hamiltonian describing the excited system through the first qubit driving line is formally written

<math>H_D/\hbar=A(t)\cos(\tilde{\omega}_2t)\left(\sigma_x \otimes I-\frac{J}{\Delta_{12}}\sigma_z\otimes\sigma_x+m_{12}I\otimes\sigma_x\right)</math>,

where <math>A(t)</math> is the shape of the microwave pulse in time, <math>\tilde{\omega}_2</math> is resonance frequency of the second qubit, <math>\{I,\sigma_x,\sigma_y,\sigma_z\}</math> are the [[Pauli matrices]], <math>J</math> is the coupling coefficient between the two qubits via the resonator, <math>\Delta_{12}\equiv \omega_1-\omega_2</math> is qubit detuning, <math>m_{12}</math> is stray (unwanted) coupling between qubits, and <math>\hbar</math> is [[Physics:Planck constant|Planck's constant]] divided by <math>2\pi</math>. The time [[Integral|integral]] over <math>A(t)</math> determines the angle of rotation. Unwanted rotations from the first and third terms of the Hamiltonian can be compensated for with single qubit operations. The remaining component, combined with single qubit rotations, forms a basis for the [[Special unitary group|su(4)]] [[Lie algebra]].

==Qubit readout==
Architecture-specific readout, or [[Physics:Quantum measurement|measurement]], mechanisms exist. Readout of a phase qubit is explained in the qubit archetypes table above. A flux qubit state is often read using an adjustable DC-[[Physics:SQUID|SQUID]] [[Physics:Magnetometer|magnetometer]]. States may also be measured using an [[Physics:Electrometer|electrometer]].<ref name="docs.pennylane.ai">{{Cite web |title=PennyLane Documentation — PennyLane |url=https://docs.pennylane.ai/en/stable/index.html |access-date=2022-12-11 |website=docs.pennylane.ai |language=en}}</ref> A more general readout scheme includes a coupling to a microwave [[Physics:Resonator|resonator]], where  resonance frequency of the resonator is dispersively shifted by the qubit state.<ref name=NatRev2017>{{cite journal |last1=Gambetta |first1=Jay M. |last2=Chow |first2=Jerry M. |last3=Steffen |first3=Matthias |title=Building logical qubits in a superconducting quantum computing system |journal=[[Physics:Npj Quantum Information|npj Quantum Information]] |date=13 January 2017 |volume=3 |issue=1 |pages=2 |doi=10.1038/s41534-016-0004-0 |doi-access=free |bibcode=2017npjQI...3....2G |arxiv=1510.04375 }}
</ref><ref name="Dispersive Readout">{{cite journal |last1=Blais |first1=Alexandre |last2=Huang |first2=Ren-Shou |last3=Wallraff |first3=Andreas |last4=Girvin |first4=Steven |last5=Schoelkopf |first5=Robert |title=Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation |journal=Phys. Rev. A |date=2004 |volume=69 |issue=6 |pages=062320 |doi=10.1103/PhysRevA.69.062320 |url=https://link.aps.org/doi/10.1103/PhysRevA.69.062320|arxiv=cond-mat/0402216 |bibcode=2004PhRvA..69f2320B |s2cid=20427333 }}</ref>

==DiVincenzo's criteria==
[[DiVincenzo's criteria]] is a list describing the requirements for a physical system to be capable of implementing a logical qubit. DiVincenzo's criteria is satisfied by superconducting quantum computing implementation. Much of the current development effort in superconducting quantum computing aim to achieve interconnect, control, and [[Engineering:Readout integrated circuit|readout]] in the 3rd dimension with additional lithography layers.The list of DiVincenzo's criteria for a physical system to implement a logical qubit is satisfied by the implementation of superconducting qubits. Although DiVincenzo's criteria as originally proposed consists of five criteria required for physically implementing a quantum computer, the more complete list consists of seven criteria as it takes into account communication over a computer network capable of transmitting quantum information between computers, known as the “quantum internet”. Therefore, the first five criteria ensure successful quantum computing, while the final two criteria allow for quantum communication.

# '''A scalable physical system with well characterized qubits.'''  "Well characterized implies that that [[Physics:Hamiltonian mechanics|Hamiltonian function]] must be well-defined i.e the energy eigenstates of the qubit should be able to be quantified.. A scalable system is self-explanatory, it indicates that this ability to regulate a qubit should be augmentable for multiple more qubits. Herein lies the major issue Quantum Computers face, as more qubits are implemented it leads to a exponential increase in cost and other physical implementations which pale in comparison to the enhanced speed it may offer.<ref name="qc-at-davis.github.io"/> As superconducting qubits are fabricated on a chip, the many-qubit system is readily scalable. Qubits are allocated on the 2D surface of the chip. The demand for well characterized qubits is fulfilled with (a) qubit non-linearity (accessing only two of the available energy levels) and (b) accessing a single qubit at a time (rather than the entire many-qubit system) by way of per-qubit dedicated control lines and/or frequency separation, or tuning out, of different qubits.
# '''Ability to initialize the state of qubits to a simple fiducial state.'''<ref name="DiVincenzo-2008">{{Cite journal |last=DiVincenzo |first=David |date=February 1, 2008 |title=The Physical Implementation of Quantum Computation |journal=IBM T.J. Watson Research Center|volume=48 |issue=9–11 |pages=771–783 |doi=10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E |arxiv=quant-ph/0002077 |bibcode=2000ForPh..48..771D |s2cid=15439711 }}</ref> A fiducial state is one that is easily and consistently replicable and is useful in quantum computing as it may be used to guarantee the initial state of qubits. One simple way to initialize a superconducting qubit is to wait long enough for the qubits to relax to the ground state. Controlling qubit potential with tuning knobs allows faster initialization mechanisms.
# '''Long relevant decoherence times'''<ref name="DiVincenzo-2008" />'''.''' Decoherence of superconducting qubits is affected by multiple factors. Most decoherence is attributed to the quality of the Josephson junction and imperfections in the chip substrate. Due to their mesoscopic scale, superconducting qubits are relatively short lived. Nevertheless, thousands of gate operations have been demonstrated in these many-qubit systems.<ref>{{cite journal |last1=Devoret |first1=M. H. |last2=Schoelkopf |first2=R. J. |title=Superconducting Circuits for Quantum Information: An Outlook |journal=Science |date=7 March 2013 |volume=339 |issue=6124 |pages=1169–1174 |doi=10.1126/science.1231930|pmid=23471399 |bibcode=2013Sci...339.1169D |s2cid=10123022 }}</ref>
# '''A “universal” set of quantum gates.'''<ref name="DiVincenzo-2008" /> Superconducting qubits allow arbitrary rotations in the Bloch sphere with pulsed microwave signals, implementing single qubit gates. <math>\sigma_z \sigma_z</math> and <math>\sigma_x \sigma_x</math> couplings are shown for most implementations and for complementing the universal gate set.<ref>{{cite journal |last1=Chow |first1=Jerry M. |last2=Gambetta |first2=Jay M. |last3=Córcoles |first3=A. D. |last4=Merkel |first4=Seth T. |last5=Smolin |first5=John A. |last6=Rigetti |first6=Chad |last7=Poletto |first7=S. |last8=Keefe |first8=George A. |last9=Rothwell |first9=Mary B. |last10=Rozen |first10=J. R. |last11=Ketchen |first11=Mark B. |last12=Steffen |first12=M. |title=Universal Quantum Gate Set Approaching Fault-Tolerant Thresholds with Superconducting Qubits |arxiv=1202.5344 |journal=Physical Review Letters |date=9 August 2012 |volume=109 |issue=6 |pages=060501 |doi=10.1103/PhysRevLett.109.060501|pmid=23006254 |bibcode=2012PhRvL.109f0501C |s2cid=39874288 }}</ref><ref>{{cite journal |last1=Niskanen |first1=A. O. |last2=Harrabi |first2=K. |last3=Yoshihara |first3=F. |last4=Nakamura |first4=Y. |last5=Lloyd |first5=S. |last6=Tsai |first6=J. S. |title=Quantum Coherent Tunable Coupling of Superconducting Qubits |journal=Science |date=4 May 2007 |volume=316 |issue=5825 |pages=723–726 |doi=10.1126/science.1141324|pmid=17478714 |bibcode=2007Sci...316..723N |s2cid=43175104 }}</ref> This criterion may also be satisfied by coupling two transmons with a coupling capacitor.<ref name="docs.pennylane.ai" /> 
# '''Qubit-specific measurement ability.'''<ref name="DiVincenzo-2008" /> In general, single superconducting qubits are used for control or for measurement.
# '''Interconvertibility of stationary and flying qubits.'''<ref name="DiVincenzo-2008" /> While stationary qubits are used to store information or perform calculations, flying qubits transmit information macroscopically. Qubits should be capable of converting from being a stationary qubit to being a flying qubit and vice versa. 
# '''Reliable transmission of flying qubits between specified locations.'''<ref name="DiVincenzo-2008" />
The final two criteria have been experimentally proven by research performed by [[Organization:ETH Zurich|ETH]] with two superconducting qubits connected by a [[Engineering:Coaxial cable|coaxial cable]].<ref>{{Cite web |last1=Morsch |first1=Oliver |last2=Zurich |first2=E. T. H. |title=Quantum transfer at the push of a button |url=https://phys.org/news/2018-06-quantum-button.html |access-date=2022-12-09 |website=phys.org |language=en}}</ref>

== Challenges ==
One of the primary challenges of superconducting quantum computing is the extremely low temperatures at which superconductors like Bose-Einstein Condensates exist. Other basic challenges in superconducting qubit design are shaping the potential well and choosing particle mass such that energy separation between two specific energy levels is unique, differing from all other interlevel energy separation in the system, since these two levels are used as logical states of the qubit.

Superconducting quantum computing must also mitigate [[Physics:Quantum noise|quantum noise]] (disruptions of the system caused by its interaction with an environment) as well as [[Leakage (machine learning)|leakage]] (information being lost to the surrounding environment). One way to reduce leakage is with [[Parity measurement|parity measurement]]s.<ref name="Kjaergaard-2020" /> Many current challenges faced by superconducting quantum computing lie in the field of microwave engineering.<ref name="NatRev2017" /> As superconducting quantum computing approaches larger scale devices, researchers face difficulties in qubit coherence, scalable [[Calibration|calibration]] software, efficient determination of [[Fidelity of quantum states|fidelity]] of quantum states across an entire chip, and qubit and gate fidelity.<ref name="Kjaergaard-2020" /> Moreover, superconducting quantum computing devices must be reliably reproducible at increasingly large scales such that they are compatible with these improvements.<ref name="Kjaergaard-2020" />

'''The Journey of Superconducting Quantum Computing:'''

Although not the newest development, the focus began to shift onto superconducting qubits in the latter half of the 1990s when quantum tunneling across Josephson junctions became apparent which allowed for the realization that quantum computing could be achieved through these superconducting qubits.<ref name="Qiskit">{{Cite web |last=Qiskit |date=2022-09-28 |title=How The First Superconducting Qubit Changed Quantum Computing Forever |url=https://medium.com/qiskit/how-the-first-superconducting-qubit-changed-quantum-computing-forever-96cf261b8498 |access-date=2022-12-13 |website=Qiskit |language=en}}</ref>

At the end of the century in 1999, a paper<ref>{{Cite journal |last1=Nakamura |first1=Y. |last2=Pashkin |first2=Yu A. |last3=Tsai |first3=J. S. |date=April 1999 |title=Coherent control of macroscopic quantum states in a single-Cooper-pair box |journal=Nature |volume=398 |issue=6730 |pages=786–788 |doi=10.1038/19718 |arxiv=cond-mat/9904003 |bibcode=1999Natur.398..786N |s2cid=4392755 |issn=0028-0836}}</ref> was published by Yasunobu Nakamura, which exhibited the initial design of a superconducting qubit which is now known as the "charge qubit". This is the primary basis point on which later designs amended upon. These initial qubits had their limitations in respect to maintaining long coherence times and destructive measurements. The further amendment to this initial breakthrough lead to the invention of the phase and flux qubit and subsequently resulting in the transmon qubit which is now widely and primarily used in Superconducting Quantum Computing.The transmon qubit has enhanced original designs and has further cushioned charge noise from the qubit.<ref name="Qiskit"/>

The journey has been long, arduous and full of breakthroughs but has seen significant advancements in the recent history and has massive potential for revolutionizing computing.

'''The Future of Superconducting Quantum Computing:'''

The sector's leading industry giants, like Google, IBM and Baidu, are using superconducting quantum computing and transmon qubits to make leaps and bounds in the area of quantum computing.

In August 2022, Baidu released its plans to build a fully integrated top to bottom quantum computer which incorporated superconducting qubits. This computer will be all encompassing with hardware, software and applications fully integrated. This is a first in the world of quantum computing and will lead to ground-breaking advancements.<ref>{{Cite press release |title=Baidu Releases Superconducting Quantum Computer and World's First All-Platform Integration Solution, Making Quantum Computing Within Reach |url=https://www.prnewswire.com/news-releases/baidu-releases-superconducting-quantum-computer-and-worlds-first-all-platform-integration-solution-making-quantum-computing-within-reach-301612321.html |access-date=2022-12-13 |website=www.prnewswire.com |language=en}}</ref>

IBM released the following roadmap publicly that they have set for their quantum computers which also incorporated superconducting qubits and the transmon qubit.

2021: In 2021, IBM came out with their 127-qubit processor.<ref name="IBM Research Blog-2021">{{Cite web |date=2021-02-09 |title=IBM Quantum roadmap to build quantum-centric supercomputers |url=https://research.ibm.com/blog/ibm-quantum-roadmap-2025 |access-date=2022-12-13 |website=IBM Research Blog |language=en}}</ref>

2022: On November 9, IBM announced its 433 qubit processor called "Osprey".<ref>{{Cite web |last=Lardinois |first=Frederic |date=2022-11-09 |title=IBM unveils its 433 qubit Osprey quantum computer |url=https://techcrunch.com/2022/11/09/ibm-unveils-its-433-qubit-osprey-quantum-computer/ |access-date=2022-12-13 |website=TechCrunch |language=en-US}}</ref>

2023: IBM plan on releasing their Condor quantum processor with 1,121 qubits.<ref name="IBM Research Blog-2021"/>

2024: IBM plan on releasing their Flamingo quantum processor with 1,386+ qubits.<ref name="IBM Research Blog-2021" />

2025: IBM plan on releasing their Kookaburra quantum processor with 4,158+ qubits.<ref name="IBM Research Blog-2021" />

2026 and beyond: IBM plan on releasing a quantum processor that scaled beyond 10,00 qubits to a 100,000 qubits.<ref name="IBM Research Blog-2021" />

Google in 2016, implemented 16 qubits to convey a demonstration of the Fermi-Hubbard Model. In another recent experiment, Google used 17 qubits to optimize the Sherrington-Kirkpatrick model. Google produced the Sycamore quantum computer which performed a task in 200 seconds that would have taken 10,000 years on a classical computer.<ref>{{Cite web |title=Our quantum computing journey |url=https://dreamcoat-guggenheim.uc.r.appspot.com/learn/map/ |access-date=2022-12-13 |website=Google Quantum AI |language=en}}</ref>

==References==
{{Reflist}}

==External links==
*[https://www.ibm.com/quantum-computing/systems/ IBM Quantum] offers access to over 20 quantum computer systems.
*The [http://www.research.ibm.com/quantum/ IBM Quantum Experience] offers free access to writing quantum algorithms and executing them on 5 qubit quantum computers.
*[https://research.ibm.com/blog/quantum-development-roadmap/ IBM's roadmap for quantum computing] shows 65 qubit systems available in 2020 and 127 qubits to be available sometime in 2021.

{{quantum computing}}

[[Category:Quantum information science]]
[[Category:Quantum electronics]]
[[Category:Superconductivity]]

{{Sourceattribution|Superconducting quantum computing}}